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Theorem ixxex 11483
Description: The set of intervals of extended reals exists. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
Hypothesis
Ref Expression
ixx.1  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
Assertion
Ref Expression
ixxex  |-  O  e. 
_V
Distinct variable groups:    x, y,
z, R    x, S, y, z
Allowed substitution hints:    O( x, y, z)

Proof of Theorem ixxex
StepHypRef Expression
1 xrex 11158 . . . 4  |-  RR*  e.  _V
21, 1xpex 6525 . . 3  |-  ( RR*  X. 
RR* )  e.  _V
31pwex 4565 . . 3  |-  ~P RR*  e.  _V
42, 3xpex 6525 . 2  |-  ( (
RR*  X.  RR* )  X. 
~P RR* )  e.  _V
5 ixx.1 . . . 4  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
65ixxf 11482 . . 3  |-  O :
( RR*  X.  RR* ) --> ~P RR*
7 fssxp 5668 . . 3  |-  ( O : ( RR*  X.  RR* )
--> ~P RR*  ->  O  C_  ( ( RR*  X.  RR* )  X.  ~P RR* )
)
86, 7ax-mp 5 . 2  |-  O  C_  ( ( RR*  X.  RR* )  X.  ~P RR* )
94, 8ssexi 4527 1  |-  O  e. 
_V
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1399    e. wcel 1836   {crab 2750   _Vcvv 3051    C_ wss 3406   ~Pcpw 3944   class class class wbr 4384    X. cxp 4928   -->wf 5509    |-> cmpt2 6220   RR*cxr 9560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-cnex 9481  ax-resscn 9482
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-fv 5521  df-oprab 6222  df-mpt2 6223  df-1st 6721  df-2nd 6722  df-xr 9565
This theorem is referenced by:  iooex  11495
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